Estimates of the convergence rate in the limit theorems on transition phenomena for branching random processes

В данной работе рассматриваются ветвящиеся случайные процессы с дискретным временем в двух предположениях: в начальный момент времени имеется одна частица или в начальный момент времени существует большое число частиц. В переходных явлениях для таких ветвящихся случайных процессов получены оценки скорости сходимости условных законов распределений к предельному распределению. We consider branching random processes with discrete time in two assumptions: at the initial moment of time there is one particle and there are large number of particles. In transition phenomena for such branching random processes, estimates of the convergence rate of conditional distributions are obtained.

Characteristic Functions

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

Conditioning

This chapter deals in depth with the concept of conditional expectation. This is defined first in the traditional “naïve” manner, and then using the measure theoretic approach. A comprehensive set of properties of the conditional expectation are proved, generalizing several results of Ch. 9, and then multiple sub‐σ‎‎‐fields and nesting are considered, concluding with a treatment of conditional distributions and conditional independence.

Forecast score distributions with imperfect observations

The field of statistics has become one of the mathematical foundations in forecast evaluation studies, especially with regard to computing scoring rules. The classical paradigm of scoring rules is to discriminate between two different forecasts by comparing them with observations. The probability distribution of the observed record is assumed to be perfect as a verification benchmark. In practice, however, observations are almost always tainted by errors and uncertainties. These may be due to homogenization problems, instrumental deficiencies, the need for indirect reconstructions from other sources (e.g., radar data), model errors in gridded products like reanalysis, or any other data-recording issues. If the yardstick used to compare forecasts is imprecise, one can wonder whether such types of errors may or may not have a strong influence on decisions based on classical scoring rules. We propose a new scoring rule scheme in the context of models that incorporate errors of the verification data. We rely on existing scoring rules and incorporate uncertainty and error of the verification data through a hidden variable and the conditional expectation of scores when they are viewed as a random variable. The proposed scoring framework is applied to standard setups, mainly an additive Gaussian noise model and a multiplicative Gamma noise model. These classical examples provide known and tractable conditional distributions and, consequently, allow us to interpret explicit expressions of our score. By considering scores to be random variables, one can access the entire range of their distribution. In particular, we illustrate that the commonly used mean score can be a misleading representative of the distribution when the latter is highly skewed or has heavy tails. In a simulation study, through the power of a statistical test, we demonstrate the ability of the newly proposed score to better discriminate between forecasts when verification data are subject to uncertainty compared with the scores used in practice. We apply the benefit of accounting for the uncertainty of the verification data in the scoring procedure on a dataset of surface wind speed from measurements and numerical model outputs. Finally, we open some discussions on the use of this proposed scoring framework for non-explicit conditional distributions.

Forecasting transaction counts with integer-valued GARCH models

Using numerous transaction data on the number of stock trades, we conduct a forecasting exercise with INGARCH models, governed by various conditional distributions; the Poisson, the linear and quadratic negative binomial, the double Poisson and the generalized Poisson. The model parameters are estimated with efficient Markov Chain Monte Carlo methods, while forecast evaluation is done by calculating point and density forecasts.

Macroeconomic and Financial Risks: A Tale of Mean and Volatility

We study the joint conditional distribution of GDP growth and corporate credit spreads using a stochastic volatility VAR. Our estimates display significant cyclical co-movement in uncertainty (the volatility implied by the conditional distributions), and risk (the probability of tail events) between the two variables. We also find that the interaction between two shocks--a main business cycle shock as in Angeletos et al. (2020) and a main financial shock--is crucial to account for the variation in uncertainty and risk, especially around crises. Our results highlight the importance of using multivariate nonlinear models to understand the determinants of uncertainty and risk.

Estimating drift and minorization coefficients for Gibbs sampling algorithms

Gibbs samplers are common Markov chain Monte Carlo (MCMC) algorithms that are used to sample from intractable probability distributions when sampling directly from full conditional distributions is possible. These types of MCMC algorithms come up frequently in many applications, and because of their popularity it is important to have a sense of how long it takes for the Gibbs sampler to become close to its stationary distribution. To this end, it is common to rely on the values of drift and minorization coefficients to bound the mixing time of the Gibbs sampler. This manuscript provides a computational method for estimating these coefficients. Herein, we detail the several advantages of the proposed methods, as well as the limitations of this approach. These limitations are primarily related to the “curse of dimensionality”, which for these methods is caused by necessary increases in the numbers of initial states from which chains need be run and the need for an exponentially increasing number of grid points for estimation of minorization coefficients.